I said above to the UK, for that you do HAVE it, no action required, it is immediate at birth, but still, if you don’t know that, yes you may be messing about with visas and rights of abode and papers for a long time if you don’t know that you can get the UK passport for travel back there. Different countries different rules, but here the father is from the UK.
A certificate of entitlement to the right of abode is given to UK citizens who for whatever reason can not hold a UK passport.
“Generally, in order to have the right of abode in a certain country, a person must be a citizen”
In this case, the original posters daughter was born in China to a Chinese parent which also makes her a Chinese citizen living in China.
If she were to obtain a UK passport, she would not be able to leave the country on it as it would not contain a visa or entry stamp for China. She could leave China on her Chinese passport but she would need a UK visa in it to do so. The UK authorities will not issue a visa to a UK passport holder. This means she can not legally obtain a visa in her Chinese passport. She could lie about holding a UK passport on the UK visa application form, but this would be a criminal offence. The best option in this situation is not to apply for a British Passport, but to apply for a certificate of entitlement to the right of abode in the Chinese passport.
Basically, yes she is a UK citizen and that is what the certificate to right of abode proves.